VALUATION OF MORTGAGE-BACKED SECURITIES

Rui Huang Bachelor of Finance, Shanghai Institute of Foreign Trade, 2004

PROJECT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF ARTS

In the Faculty

of Business Administration

Financial Risk Management

O Rui Huang 2006

SIMON FRASER UNIVERSITY

Summer 2006

All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without permission of the author.

APPROVAL

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Title of Project:

Rui Huang

Master of Arts

Valuation of Mortgage-backed Securities

Supervisory Committee:

Geoffrey Poitras Senior Supervisor Professor of Business Administration

Date Approved:

Chris Veld Supervisor Associate Professor of Business Administration

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ABSTRACT

This paper attempts to provide a method for the valuation of mortgage-backed securities

(MBS).

Specifically, Modified Goldman Sachs model is selected to describe mortgagors'

prepayment behaviour, which takes account of mortgage's refinancing incentive, aging effect,

month effect and burnout effect. In this study, I will use above model to price a pass-through

MBS and a plain vanilla MBS respectively.

Furthermore, I use scenarios testing to discuss how MBS's price and standard deviation

(risk) change if mortgage property and model parameters change.

Keywords: Mortgage-backed Securities; Prepayment Model; Morte Carlo Simulation

ACKNOWLEDGEMENTS

I would like to thank Professor Geoffiey Poitras for his encouragement and continued

support for my paper.

I also wish to thank Professor Robbie Jones for providing his valuable treasure, sharing

his knowledge with me.

TABLE OF CONTENTS

. . Approval .......................................................................................................................................... 11

... Abstract .................................................................................................................................. 111

Acknowledgements ........................................................................................................................ iv

........................................................................................................................... Table of Contents v ..

List of Figures ............................................................................................................................... VII

.. ................................................................................................................................ List of Tables VII

.............................................................................................................................. Introduction 1

...................................................................................... Risk of Mortgage-backed Securities 2 ..................................................................................................................... 2.1 Refinancing 2

...................................................................................... 2.2 House Turnover or House Sales 2 .................................................................................................. 2.3 Involuntary Prepayment 2

Mortgage Modelling Overview ............................................................................................... 4 .............................................................................................................. 3.1 Structural Model 4

..................................................................................................... 3.2 Reduced Form Model 5

Prepayment Model ................................................................................................................... 7 ...................................................................................................... 4.1 12-year Life Method 7

4.2 FHA Experience Method ................................................................................................ 7 ................................................................. 4.3 CPR (Conditional Prepayment Rate) Method 7

............................................... 4.3.1 PSA (Public Security Association) Standard Method 8 ..................................... 4.3.2 Schwartz and Torous 's Proportional Hazard Model (PHM) 8

................................................ 4.3.3 Richard and Roll's Modified Goldman Sachs Model 9

............................................................................................................... Interest Rate Model 12

............................................................................................................... 5.1 Vasicek Model 12 ..................................................................................................................... 5.2 CIR Model 12

Plain Vanilla MBS ............................................................................................................. 14

MBS Cash Flows .................................................................................................................... 15

........................................................................................................................... MBS Pricing 19 ................................................................................................ 8.1 Pricing a Mortgage Pool 19 ............................................................................................... 8.1.1 Change Coupon Rate 25

........................................................................................... 8.1.2 Change Mortgage Term 26

.......................................................................................... 8.1.3 Change CIR Parameters -26

......................................................................................... 8.2 Pricing a Plain Vanilla MBS 28 8.2.1 Change Mortgage Term ........................................................................................... 29

.......................................................................................... 8.2.2 Change CIR Parameters 31

.............................................................................................................................. 9 Conclusion 33

Reference List ............................................................................................................................... 34

LIST OF FIGURES

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Month Factor .............................................................................................................. 11

Cash Flow Pattern for Regular Bonds ................................................................... 15

Cash Flow Pattern for MBS ..................................................................................... 16

Computation Process .............................................................................................. 19

One Interest Rate Path ................................................................................................ 21

One R Path .................................................................................................................. 23

One CPR Path ............................................................................................................ 2 4

............................................. Positive Relationship Between Price and Coupon Rate 26

LIST OF TABLES

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6

Table 7

Table 8

Table 9

Mortgage Pool ............................................................................................................ 20

Estimated Results for Mortgage Pool (Different Coupon Rate) ................................. 25

Estimated Results for Mortgage Pool (Different Mortgage Term) ............................. 26

Estimated Results for Mortgage Pool (Different Mean Reverse Speed) .................... 27

Estimated Results for Mortgage Pool (Different Interest Volatility) .......................... 27

CMO Structure ........................................................................................................... 28

Tranche Price (Different Mortgage Term) .................................................................. 29

Tranche Price (Different Mean Reverse Speed) ........................................................ 3 1

Tranche Price (Different Interest Volatility) .............................................................. 32

vii

INTRODUCTION

Mortgage-backed securities (MBS) are investment instruments that represent ownership

of a group of mortgages. Usually, financial institutions treat MBS as a method to fund the capital

pool brought by lending mortgages.

Up to 2006, MBS market is one of the fastest growing, as well as one of the largest

financial markets in the United States. There are now three major agencies in U.S. issuing MBS -

Ginnie Mae (the Government National Mortgage Association, or GNMA), Fannie Mae (the

Federal National Mortgage Association, or FNMA), and Freddie Mac (the Federal Home Loan

Mortgage Corporation, or FHLMC). These agencies support the secondary market by issuing an

MBS in exchange for pools of mortgages from lenders. This ensures that lenders have funds to

make additional loans. In general, MBS issued through these three government-supported

agencies are considered to have no credit or negligible credit risk.

The paper will be structured in this way. In section 2, we will look at risks behind

mortgage-backed securities and factors causing those risks. Mortgage pricing models developed

in the recent past will be presented in section 3. In section 4, we will discuss some prepayment

models and Modified Goldman Sachs model is chosen to price MBS. In section 5, we will choose

the interest rate process. Then in section 6, we will introduce the specific CMO structure we will

price later - plain vanilla. In the next section, section 7, MBS cash flow streams are discussed. In

section 8, the computation process is detailed, and we will present the numerical results. Section 9

will be the last section where the conclusion is presented.

2 RISK OF MORTGAGE-BACKED SECURITIES

The biggest uncertainty of mortgage-backed securities is the prepayment behaviour. In

U.S., mortgagors could prepay their loans at any time with no or only little penalty. The above

earlier payments lead to unstable fiture cash flow, thus uncertainty of MBS fair price today. To

some extent, mortgages can be treated as a callable bond and people could exercise their option

any time they like. So before we begin to price MBS, What we want to know is at what time

those borrowers will exercise the call option and what the vaiue of that embedded option is.

Roughly, there are three reasons causing the prepayment behaviours.

2.1 Refinancing

When interest rates decline, mortgagors originally signed the higher rate contract would

like to refinance through entering a new mortgage contract, thus to take advantage of the

prevailing low interest rates.

2.2 House Turnover or House Sales

Many mortgage contracts specify the settlement of remaining loan before selling out the

house. So people would like to prepay before mortgage mahrity date if they want to sell the

house.

2.3 Involuntary Prepayment

Prepayment could occur unexpectedly. Such early termination events include marriage,

divorce, death and other extraordinary events.

Lots of studies then report that several factors affect those behaviours - interest rate as

mentioned above, mortgage seasoning, borrowers' heteroscedasticity, overall economy, burnout

effect, even geographic diversity and schooling time.. . . It seems to be a tough work if we want to

construct a model with these multitudinous factors, but still, several models have been built

through chasing one or several above variables.

3 MORTGAGE MODELLING OVERVIEW

Many literatures on mortgage valuation have been published in the past 20 years.

3.1 Structural Model

Kenneth Dunn was the first to introduce an MBS model and then a series of the papers

with McConnell in 198 1. They treat prepayment as a call option and their model assumes optimal

refinancing decisions. Although these models consistently link valuation and prepayment, their

prepayment predictions assume an instant prepayment action by all the mortgagors when interest

rates hit a threshold. Tirnmis, Dunn and Spatt, and Johnston and Van Drunen improved this

model by incorporating transaction cost or other frictions to explain refinancing decisions.

Noticing that some people may not exercise their options when interest rate is low, in the

following years, this option-based method has been improved by adding so called option

probability model or other determined variables. Downing, Stanton and Wallance established a 2

factor model use both interest rate and house price as a joint threshold and option probability

model by a hazard rate fimction. Other researchers, like Kariya, Ushiyama, Pliska construct a 3

factor model using treasury yield, mortgage rate and house price.

The general idea of the structural model begins with the "public believed" movement of

determined variables, usually interest rate and house price. Then Ito's lemma is used to get a

"pure" PDE equation and prepayment noises are added to compute the "real" PDE equation.

Finally we solve this PDE equation with boundary condition.

For example, in 2 factor model, where we assume house price and interest move as

follows.

dH = (u, - s)Hdt + o, H ~ Z , ~

Where s represents risk adjusted drift when we applied to the risk neutral world.

dr = y(B - r)dt + or &dzr CIR (1985)

We could get PDE for mortgage X(r,H,t) (That is the core idea of the structural model)

Combining this pure equation with boundary condition and option exercise probability

model, we could get the final numerical answer. Today, different finite backward methods are

used to solve this problem.

It seems to be a compromise of reality, since option-base technique is a "perfect,

classical, non-arbitrage" thought that investors act optimally, while option exercise probability

model is generally a result of irrational exercise behaviours.

3.2 Reduced Form Model

Since above theoretical process asks for very complex calculation, practitioners in

industry refer to find other comparatively easy-to-understand solution. Reduced form model

assumes option exercise is built in a prepayment model and evaluates mortgage price by

discounting future cash flow to its present value. It is easy to modify and needs less computation.

No doubt when those academic geniuses were hired by those big investing companies or financial

institutions they all choose to use this method.

The basic idea of this method is that the price of MBS today should be the present value

of total cash inflow which could be earned in the future.

Here we assume a 30-year mortgage, v l , v2,. . . are explainable variables, 21, 22,. . . are

implied yield.

Usually, cash flows are determined by a prepayment model and Monte Carlo simulation

is used to simulate interest process and cash flows. In the paper we prefer this method and will

use it to compute MBS pool, but first, let us look at some different prepayment models.

4 PREPAYMENT MODEL

4.1 12-year Life Method

According to the material from Federal Housing Administration (FHA), a 30-year

mortgage loan will be hlly paid in the 12th year, so GNMA obtain MBS price by assuming that

the 30-year mortgage will have normal payment in the first 143rd months then repay the all the

balance remaining in the 144th month. This coarse method is quickly substituted by other more

complex method since there is evidence that the prepayment model is determined by a number of

different factors.

4.2 FHA Experience Method

FHA built a yearly survivorship table according to its historical 30-year mortgage data.

Prepayment rate is lower during initial loan period, then increases, and reaches its crest between

the 5th and 8th year, afterwards the number drops. The advantage of this method is prepayment

could vary every year. The disadvantage is that those numbers only reflect the prepayment rate in

a specified period and do not take interest rate into account.

4.3 CPR (Conditional Prepayment Rate) Method

There has been a proliferation of CPR models in financial literatures. These models

include those by PSA (Public Security Association), Asay, Guillaurne and Mattu (1987), Chinloy

(1 989), Davidson, Herskovitz and Van Drunen (1 989), Giliberto and Thibodeau (1 989), Richard

and Roll (1989), Schwartz and Torous (1989). Among those models, PSA method is an "easy to

implement" one and a learning objective in recent CFA curriculum. The last two models are more

popular and referenced time after time in the recent papers, so we will discuss them in detailed.

4.3.1 PSA (Public Security Association) Standard Method

Public Security Association supposes the prepayment rate in the first year is 0.2% and

this rate grows by 0.2% every month until the 3oth month. It could be formulated by:

t CPR =6%*- when t<30

30

CPR = 6% when t>=30

PSA treats above formula as its 100% mortgage prepayment benchmark. For different

PSA mortgage, for example, 150% PSA, you just need to multiply 1.5 on the above formula.

4.3.2 Schwartz and Torous's Proportional Hazard Model (PHM)

They use the following proportional hazard model to compute CPR.

CPR = ~ ( t ; v, ,8) = r,, (t; r, p) * exp(pv)

v is the vector of determined factors; P is the vector of coefficients.

Particularly, they use the following four factors.

Here v, represents the difference between mortgage rate and current long term treasury

rate. c is mortgage rate; 1 is long term treasury rate at time t-s, where s is the lag factor. s=3 is

used in this model.

Here v2 is just cubed equation of vl, representing the accelerate effect when treasury rate

is sufficiently lower than the mortgage rate.

v, (t) = ln(BAL(t) l BAL * (t))

v3 represents the burnout effect where BAL(t) is the mortgage pool outstanding at time t

and BAL*(~) is pool outstanding in the absence of prepayment.

1 when t = May through August

0 when t = September through April

v4 represents the seasoning factor on prepayment decision.

zo (t; r, p) is baseline function which explain the CPR when factor vector v=O, and they

use log-logistic function to describe .no

This model is intuitive because it divides the mortgage prepayment risk into two parts.

The first part, 7~ (also called aging effect) tells us that mortgage will suffer a baseline risk at any

time of mortgage period. The second part, exp (pv) represents all the mortgage specified risk.

4.3.3 Richard and Roll's Modified Goldman Sachs Model

The model is first constructed when Richard and Roll were hired by Goldman Sachs as

financial advisors and then developed by OTS (Office of thrift supervision). They point out that

CPR is determined by the following four factors and measure CPR as the product of those four

factors:

CPR = (Refinancing Incentive)*(Seasoning Factor)*(Month Factor)*(Pool Burnout Factor)

Where CPR is the conditional annual prepayment rate.

Refinancing incentive in their model could be measured by the ratio or the spread of

weighted average mortgage coupon rate and mortgage refinancing rate.

RI = a + b * arctan(c + d(WAC - R))

Seasoning reflects the observation that newer loans tend to prepay slower than older or

"seasoned" loans. This factor follows the rationale behind the PSA standard prepayment model.

This industry convention adopted by the Public Securities Association models mortgage

prepayment rates as increasing linearly from 0.2% CPR at issue to 6% CPR at thirty months and

then remaining constant. In the paper, the formula for Seasoning is:

Month factor measures different prepayment behaviours during 1 year. It is believed that

prepayments peak in the summer and decrease in the winter. One of the sources of prepayments

due to seasonality is housing turnover. This could be due to weather and school schedules. In this

paper, the monthly parameters for ith month were taken from the figure in Richard and Roll's

paper (1989).

Figure 1 Month Factor

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Source: data from Richard and Roll, 1989

Burnout takes into account the tendency for prepayment to diminish over time, even

when refinancing incentives are favourable. Richard and Roll try to quantify burnout by

measuring how much the option to prepay has been deep in-the-money since the pool was issued.

They suggest the more the prepayment option has been deep in-the-money, the more burned out

the pool is, and the smaller prepayments are, all other things being equal. The Burnout factor is

calculated as follows:

Where B(t) is the mortgage balance in the beginning of month t, B(0) is the initial

mortgage balance.

5 INTEREST RATE MODEL

Interest rate is notoriously difficult to predict. Recent interest model could be categorized

into two groups - equilibrium models and non-arbitrage model. The main difference of the two

lies in the different description of interest rate drift. In equilibrium model, drift is not set as a

fknction of time while non-arbitrage model takes it into conqideration. In the paper we will

introduce non-arbitrage models which are more commonly used in MBS evaluation.

5.1 Vasicek Model

Vasicek (1 977) proposes a kind of interest rate model

dr = a(b - r)dt + a l z (Vasicek Model)

Where a is instantaneous drift speed, b is long term equilibrium interest rate, o is the

instantaneous standard deviation and dz is a Wiener process

The Advantage of this model is the built-in mean-reversion idea in the equation. When

short-term interest rate deviates from long time equilibrium level b, it will then be reversed to b at

the speed of a.

5.2 CIR Model

Cox, Ingersoll and Ross (1985) improve Vasicek's model by modifying instantaneous

standard deviation from o to 06

(CIR Model)

This model follows the idea of mean-reversion phenomenon of interest rate. Besides, it

will not generate negative interest rate. So I pick this model to build our interest rate path.

Parameters in our interest rate model are described in the "M~rtgage Pool Pricing" section.

6 PLAIN VANILLA MBS

Mortgage pool is not expected to be sold in one pile. Usually, those agencies will issue

securities by redirecting the cash flows of mortgages in order to attract different investors. They

redirect cash flows in such way that one classes of bond will be only exposed to one form of

prepayment risk. When the cash flows of mortgage-backed products are redistributed to different

bond classes, the resulting securities are called collateralized mortgage obligations (CMOs). Plain

Vanilla CMOs, also called Sequential Pay CMOs, is the most basic class within CMOS structure.

This class reallocates collateral principal payments sequentially to a series of bonds, namely

A,B,C,D.. . . Bond A will receive principal payments, including prepayments first. When it is fully

retired; then principal payments are redirected to the next sequential class, bond B. This process

continues until the last sequential class is fully retired. While one class is receiving principal

payments, the other existing classes only receive monthly interest payments at their coupon rate

on their principal.

7 MBS CASH FLOWS

As mentioned above, mortgage is just like an ordinary coupon bond (without final

principal payment) if there is no prepayment. The interest part and principal part in every

schedule payment under no prepayment assumption will follow the pattern in figure 2.

Figure 2 Cash Flow Pattern for Regular Bonds

I I

0 10 2 0 30 years

Source: www.riskglossaly.com

But when prepayments happen, the interest part and the principal part will be uncertain. It

may follow the pattern below.

Figure 3 Cash Flow Pattern for MBS

servicing fee

principal

1 interest

years

Source: www.riskglossay.com

In Fabozzi's paper, the cash flows can be calculated using the following formula:

Where MP(t) is the scheduled mortgage payment for period t, B(t) is the mortgage

balance remaining at t, WAC is weighted average coupon rate and WAM is the weighted average

maturity of mortgage.

WAC IP(t) = B(t) * -

12

Where P(t) is the interest payment at t.

Where SIJ(t) is the scheduled principal payment for period t;

PP(t) = SMM ( t ) * (B( t ) - SP(t))

Where PP(t) is the principal prepayment for period t and SMM is single month mortality

and computed as

SMM ( t ) = 1 - d m

We could calculate cash flow using CPR(t), the conditional prepayment rate, which was

derived from our prepayment model. Once CPR(t) is known, everything else can be calculated

and total cash flow at time t CF(t) will equal:

Then we could compute the price of mortgage pool hy discounting monthly cash flow.

P, = CF(1) + CF(2) + ... + CF (360) (1 + S, + OAS) ( 1 + S, + O A S ) ~ (1 + s,,, + 0 ~ s ) ~ ~ ~

Where S is the spot yield curve, computed by our simulated interest rate process.

S,, = d ( l + r l ) ( l + r 2 ) ...( l + r n ) -1

OAS is the implied spread from historic MBS data that makes the computed price of

mortgage pool equal to its market price. In this paper OAS is set to zero, since we only want to

compare pure outcomes between different classes.

The only difference between an entire mortgage pool and a Plain Vanilla MBS is that all

the principal payments and prepayments will be used to retire first sequential bonds, say Bond A.

The principal of Bond B will begin to decline until Bond A is fully amortized, so the calculation

will be as follows if we assume there are two classes, Bond A and Bond B.

W I ) + CF(2) + ... + CF ( t 1) PBondA = ( l + s , +OAS) (1+s2 +OAS)* ( 1 + s,, + 0 ~ s ) ' ~

Where t l is the time Bond A is fully retired and cash flow CF(t) will be

Where IPA(t) is the interest payment according to the remaining balance of bond A at

time t, SPT(t) and PPT(t) is the schedule principal payment and prepayment computed from the

total mortgage pool.

IP + IP + ... + CF (t 1) PBondB = + ... + CF(4

(1 + S, + OAS) (1 + S, + OAS)' (1 + sf, + OAS)" ( 1 + st + OAS)'

where IP is fixed interest payment of bond B when bond A is in the process of retiring.

CF(t1) = SPT(t1) + PPT(t1) + IPB(t1) is the first cash flow of bond B that contains

prepayment and schedule principal payment.

8 MBS PRICING

After every computation processes are introduced, we could then price MBS through the

following flow chart.

Figure 4 Computation Process

Interest Process L-h Prepayment Model c Cash Flow Stream -

MBS Price I 8.1 Pricing a Mortgage Pool

In this paper, I assume a mortgage-backed security with collateral value equal to

1000000. Its principal values, coupon rates, issue date, maturity are described as follows

(assuming Gross coupon rate = Net coupon rate, Settle date = Issue date).

Table 1 Mortgage Pool

Principal

First, I use CIR model to generate interest rate process. Particularly, I use the

discretization form of CIR model, an AR(1) model to generate interest rate path.

$1,000,000

Above AR(1) model is said to have the same mean and stationary variance if

Issue Date

2 2 w2 ( = e-" and C T ~ = CT - 2a

Jan 01, 2000

The long-run average, b, was set to 7.75%. a was set to 0.1 and the volatility term, o, was

set to 0.0225. These parameters are picked from Buser and Hendershott's article (1984). The

starting interest rate was set to 10%. Figure 5 plots one path of the interest rate process used in

my simulation.

Mortgage Term Coupon Rate

30 year 8%

Figure 5 One Interest Rate Path

Here I choose Modified Goldman Sachs prepayment model to valuate MBS. Particularly,

refinancing incentive is computed using following formula and parameters are picked from

Akesson and Lehoczky's simulation paper (2000).

RT = a + b * arctan(c + d(WAC - R))

WAC is weight average coupon rate

R(t,T) is calculated using following method.

According to Hull (2000), if interest rate follows CIR movement,

dr = a(b - r)dt + o&dz

then bond price P(t,T) will equal to

P ( ~ , T ) = e-R(tJ ')*(T-t) = A ( ~ , T ) ~ - B ( ~ . T ) * ~

2(e7'(T-t) - 1) where B(t, T) =

( y + a)(eY(T-" - 1 ) + 2 y

with y = J a 2 + 20'

Figure 6 plots one path of the R process used in my simulation.

Figure 6 One R Path

0.085

Recall that RI = a + b * arctan(c + d(WAC - R)) and CPR = (Refinancing

Incentive)*(Seasoning Factor)*(Month Factor)*(Pool Burnout Factor)

Figure 7 then plots one path of the CPR process used in my simulation.

Figure 7 One CPR Path

I ran 1000 simulations to get the MBS price. Making it simple, I set the market price of

the collateral and the bonds to their respective present values, assuming OAS to equal 0 and

observe what will happen if we change the parameters of the model.

8.1.1 Change Coupon Rate

Table 2 Estimated Results for Mortgage Pool (Different Coupon Rate)

Mortgage Pool 1

Table 2 above presents the price and standard deviation of the mortgage pool. Here I set

initial interest rate to 5%, 8% and lo%, one higher than, one lower than and one close to the long-

term rate and pick different coupon rate (4%, 6%, 8%, 10% and 12%). As coupon rate increases,

prepayment becomes more probable, thus the price of mortgage pool increase. Compared to

initial interest rate lo%, pool prices at 5% interest rate level zxe generally higher, reflecting a

more frequent prepayment behaviour in the beginning period when interest rates have not yet

converged to the long term rate, 7.75%. Figure 8 below presents the relationship between pool

price and coupon rate at two initial interest level (5% and 10%).

Coupon rate

Pool Std

4%

804,809

784,392

-

Pool Price r(0)=5%

r(0)=8%

r(0)=5%

r(0)=8%

6%

927,047

904,781

26,203

25,597

8%

1,054,560

1 ,O3U,i'2l

27,241

25,969

10%

1 ,I 88,868

1 , I 64,248

12%

1,325,394

1,298,989

27,707

28,395

27,684

27,883

27,633

28,724

Figure 8 Positive Relationship Between Price and Coupon Rate

I

Coupon ra te

8.1.2 Change Mortgage Term

Table 3 Estimated Results for Mortgage Pool (Different Mortgage Term)

Table 3 above presents the price and standard deviation of the mortgage pool in different

Mortgage Pool with coupon rate 8%

mortgage term. We could see from the table that both price and standard deviation accrue when

we increase the mortgage term. It can be indicated that uncertainty of prepayment increases when

Mortgage Term

the mortgage term is much longer.

10

1,038,228

1,016,757

1,002,264

17,701

Pool Price

Pool Std

8.1.3 Change CIR Parameters

r(0)=5%

r(0)=8%

r(0)=10%

r(0)=5%

Interest rate played an important role when we valuate the mortgage-backed securities.

15

1,044,449

1,022,209

1,008,403

21,466

Here I present the MBS results corresponding to different mean reverse speed (a=0.04, a=0.07,

a=0.1, a=0.13, a=0.16) and different interest rate volatility (a=0.0025, ~ ~ 0 . 0 1 2 5 , ~ 0 . 0 2 2 5 ,

20

1,048,359

1,026,044

1,011,768

22,594

0=0.0325). The results are shown in table 4 and table 5 respectively.

25

1,050,507

1,029,940

1,013,658

24,728

30

1,054,560

1,032,925

1,016,203

25,885

Table 4 Estimated Results for Mortgage Pool (Different Mean Reverse Speed)

Mortgage Pool with coupon rate 8%

Mean Reverse Speed 1 0.04 1 0.07 1 0.10 1 0.13 1 0.16

Pool Price

Table

Pool Std

Estimated Results for Mortgage Pool (Different Interest Volatility)

Mortgage Pool with coupon rate 8%

r(0)=5%

r(0)=10%

I Interest Volatility 1 0.0025 1 0.0125 1 0.0225 r0.0325 1

r(0)=5%

r(0)=10%

I Pool Price I r(0)=5% 1 1,056,620 1 1,055,040 1 1,054,560 1 1,051,589 1

1,059,097

988,296

I Pool Std ( r(0)=5% 1 3,107 1 15.625 1 27,707 ( 35,949 1

41,170

50,562

It seems CIR parameters have a huge effect on the pool standard deviation. Pool standard

deviation soars if we decrease mean reverse speed and increase interest volatility. What is worth

mentioning is that, when initial rate is lo%, pool prices increase with the increase of mean

reverse speed; It changes in opposite direction when initial rate is set to 5%. It could be explained

by the intuition that, at a lower initial rate, more prepayment behaviours are expected at lower

mean reverse speed than at higher reverse speed, it will increase the cash flows in the first several

years, thus increase the pool price. On the other hand, at a higher rate, more prepayment

behaviours are expected under higher mean reverse speed, so the pool price will be higher.

1,056,553

1,007,704

31,736

34,147

1,054,560

1,016,203

27,707

27,903

1,050,852

1,020,210

1,049,826

1,023,411

21,885

22,183

18,602

18,239

8.2 Pricing a Plain Vanilla MBS

After we analyze the mortgage pool, let us look at a simple CMO structure. Table 6

below presents a four tranche Plain Vanilla MBS. Coupon rates in every tranche are picked to

make the sum of tranche coupon payments equal to pool coupon payments.

Table 6 CMO Structure

Principal Issue Date Mortgage 1 Term

1000000 Jan 01, 30 year 2000

- Tranche 350000 A

Tranche 150000 D

I Total Coupon Payment

Coupon Rate

Coupon Payment

80000

The main difference is we now have interest payment for every tranche and monthly

principle payments are paid in sequence. Let IP(t)i denote the interest payments for tranche i in

month t.

WA Ci IP(t) = Bi ( t ) * -

12

I ran 500 simulations to get tranche prices and I will repeat the similar process (set initial

rate to 5% and 10%) to analyze the prices and risks under different situations.

8.2.1 Change Mortgage Term

Table 7 Tranche Price (Different Mortgage Term)

Mortgage Term

Tranche Price

Tranche Std

Plain Vanilla MBS

10 15 20 25 30

Tranche A 359,087 360,544 361,634 362,935 363,449

Tranche B 1 213,692 1 216,450 1 218,640 1 220,795 1 221,831

Tranche C 300,735 299,524 297,814 297,235 296,145

Tranche D 158,501 160,278 161,276 161,994 162,407

Tranche A 1 348,792 1 349,485 1 350,247 1 350,328 1 351,102

Tranche B

Tranche C

Tranche D

Total

I I I I I

Tranche A 1 2,530 1 3,427 1 4,121 1 4,471 1 5,064

Tranche B 1 3,111 1 3,713 1 4,447 1 4,832 1 5,543

Tranche C 1 8,399 1 9,632 1 10,566 1 10,701 1 10.933

Tranche D 1 4,975 1 5,596 1 6,035 ( 6,230 1 6,262

Total 17,554 20,665 23,520 24,388 25,978

Tranche A

Tranche B

Tranche C

Tranche D

Total

Table 7 above presents the tranche prices in different mortgage term. As expected,

tranche A has smaller standard deviation if we take principal magnitude into account. It can be

explain by the fact that senior tranches are retired at earlier period when interest rate is still away

from the long term equilibrium rate, so people are pretty sure whether they should prepay or not.

Surprisingly, all tranche prices are growing with the increase of mortgage term except tranche C.

Besides, tranche C has the largest standard deviation, almost twice as large as tranche A. These

findings may reveal the fact that tranche C's payments are in the most unstable period and many

elements impact the price of that tranche.

8.2.2 Change CIR Parameters

Table 8 Tranche Price (Different Mean Reverse Speed)

Tranche Price

Tranche Std

Plain Vanilla MBS

Mean Reverse Speed

r(0)=5%

Tranche A

Tranche B

TrancheC

Tranche D

Total

r(0)=10%

0.04

363,011

218,311

304,673

166,310

1,052,304

Tranche I3

Tranche C

Tranche D

Total

0.07

363,577

220,967

299,594

163,878

1,048,Ol 5

9,979

22,699

13,267

53,384

0.1 6

361,508

221,071

293,784

161,248

1,037,611

0.1

363,449

221,831

296,145

162,407

1,043,832

7,142

15,466

8,880

36,844

0.13

362,738

221 ,998

295,061

161,892

1,041,689

5,601

10,395

6,027

26,041

4,702

8,537

4,952

21,570

4,190

7,080

3,997

18,453

Table 9 Tranche Price (Different Interest Volatility)

Plain Vanilla MBS

nterest Volatility I 1 0.0025 1 0.0125 1 0.0225 1 0.0325

Tranche Price

Tranche A 1 364,005 1 363,813 1 363.449 1 361,758

I Tranche B 1 222,955 1 222,762 1 221,831 1 219,340 --

r(0)=5% I Tranche C 1 295,591 1 295,899 1 296.145 1 297,417

I Tranche D ( 162,255 1 162,360 1 162.407 ( 162,745

I Total 1 1,044,806 1 1,044,834 1 1.043,832 1 1,041,260

I Tranche A 1 351,434 1 351,310 1 35lIlO2 1 350,454

r(0)=10% I Tranche C 1 284,384 1 284,339 1 284.698 1 286,324

Tranche €3 21 4,749

Tranche D

Total

Tranche A

Tranche B

Tranche Std

21 4,458

156,120

1,006,688

r(0)=5%

600

654

r(O)=lO% I Tranche C 1 1,214 1 5,926 1 10,395 1 14,939

21 3,627

156,034

1,006,141

Tranche C

Tranche D

Total

Tranche A

Tranche B

212,184

2,994

3,274

Table 8 and 9 examine the CIR parameter effect on the tranche price and standard

deviation. Compared to junior tranches, senior tranches are less influenced by the change of

interest volatility and mean reverse speed. Their standard deviations change slower than the

junior ones which accord with the thought that senior tranches have comparatively stable cash

flow streams.

156,239

1,005,666

1,221

716

2,975

719

707

Tranche D

Total

156,474

1,005,435

5,064

5,543

7,069

6,840

6,109

3,492

14,846

3,505

3,404

694

3,138

10,933

6,262

25,978

5,961

5,601

3,361

15,208

14,530

8,292

33,612

8,261

7,335

6,027

26,041

8,603

36,211

CONCLUSION

Our paper provides a big picture of mortgage-backed securities and a general idea of how

to price mortgage-backed securities. In the paper, we use reduced form method to pricing MBS

and CMOS. Interest rates were simulated by using CIR model. A set of cash flow was generated

by the prepayment model and final price is computed by discounting all the cash flows. The

numerical results presented help us determine how risk is allocated in mortgage pool and each

tranches.

The results reveal that the price of mortgage pool is unconditionally positive related with

mortgage term and coupon rate and negative related with interest volatility. When we change the

interest reverse speed, the pool price gives us two different tendencies. It goes up with the

increase in the reverse speed, if we set initial rate higher than mortgage coupon rate and falls if

we set initial rate lower than mortgage coupon rate. Mortgage's standard deviation goes up with

the increase of mortgage term, coupon rate and interest volatility. It rocket even we change

volatility by 1 percent. Its risk declines if we increase the interest reverses speed, which could be

explained by the more predictable cash flows.

Things become more complex when we deal with a plain vanilla structure MBS. The

numbers still realize some tendencies of the change of the tra~che price and tranche standard

deviation. However, those tables reveal that risks have been redistributed in tranches in some

forms and could be studied by changing the coefficients. In this paper, I only assume a plain

vanilla structure; however, my simulations can be easily implemented to any other type of CMO

and more complicated CMO structures could be studied by this method.

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